I voted Yes but I don't like loot lockers findable in arena. What you bring is what you use. And it could be player regulated. Don't want to fight someone overpowered, then just find some lower-level person wanting some fun.
The arenas could be in places like junktown where there already rings, but this would be maybe a room under some buildings for quick 1 on 1, or a special area outside of Klamath for team on team matches.
Then again, if it's player-regulated expect a fuckton of griefers and spoilsports. So maybe some sort of "teleport" for players who go inside a special room and click a button in terminal saying "let's fight in this mode" and gives them an arena just for themselves.
Spectators are a bad idea. Just Fraps it. And rewards would be loot from others and whatever else was promised. For people you trust you can make friendly bets and be sure they'll pay up.
Reputation does not matter if you can respawn without any consequences 50 seconds later. That's what ruins fallout and we've no way of avoiding it. Who cares about anything if you can't die?
Well I like your idea about what you bring is what you use. But remember, that FOnline is pretty flat right now and if such arenas should be rather tactical, then something like looting at least stimpacks or some weaker drugs from already spawned corpses could provide some fun. But this depends on arena type and looting stuff should be reserved for bigger fights (Assault/CTF).
Edit: Fresh idea comes to me while I was typing this. What if looting new stuff is dependent on objectives you have successfully finished in arena. I mean for Assault it could be like one team defends other attacks. So if defending team can hold attackers for 3 minutes without loosing their first line of defense, then defenders will get some weapons/armors spawned at their respawn point. And vice versa.
Anyway in these arenas people should not be punished for dieing. At least not by loosing their items. Maybe some other punishment could be set but I think if there are some conditions to enter arenas then they should be enough. So for example you can take part in Assault only if you have suitable gear = NPC will check your armor/weapon/skills and if you have low gear/skill levels then he will ignore you
as you can't be fun to watch in arena since you would die instantly (Edit: I remembered now Assault should be something like story based so it could be done like skirmish between two factions like NCR/raiders so organizer NPC would turn you down as you are not prepared for such battle). These conditions imho should do the trick. So you bring only your weapon/armor/ammo and only drugs and some secondary weaponry (grenades, traps if implemented etc.) should be lootable. And lets say that if you loose in arena you will be punished by item detoriation and ammo that you lost while shooting other people and winner would get his ammo back + some cash maybe or something like that.
Also these arenas should NOT be accessible through regular ways. Only way how to enter them should be chat with NPC responsible for their organization. Once enough of players is signed in for battle they get memo (via radio or something) and they will be somehow transported to battlefield. When the battle is over players spawn at arena guy NPC.